
Dense passage retriever (DPR; [1]}) uses a single [CLS] vector to represent each query and document.
This is equivalent to setting \(A_{1,1}=1\)  and 0 otherwise, resulting in \(||{\mathbf {A}}||_0 = 1\) .

ME-BERT [2]} uses the first \(k\)  document token vectors for multi-vector representations of documents but a single vector for query. The similarity function is \(\max _{j=1\dots k} {\mathbf {q}}_1^\top {\mathbf {d}}_j\) , which is equivalent to setting \(A_{1,j}=1\)  when \({\mathbf {S}}_{1,j}\)  is the maximum within \({\mathbf {S}}_{1,1}\)  to \({\mathbf {S}}_{1,k}\) , and 0 otherwise. The alignment sparsity is \(||{\mathbf {A}}||_0=1\) .

ColBERT uses the sum-of-max similarity function \(\sum _{i=1}^n\max _{j=1\dots m}{\mathbf {S}}_{i,j}\)  that is equivalent to setting an alignment matrix to select the maximum element from each row of \({\mathbf {S}}\) , i.e., \({\mathbf {A}}_{i,j}=1\)  when \({\mathbf {S}}_{i,j}\)  is the maximum within \({\mathbf {S}}_{i,:}\) .
\(||{\mathbf {A}}||_0=n\)  in this case.

COIL [3]}, similar to ColBERT, also selects the maximum element from each row of \({\mathbf {S}}\) , but requires a lexical exact match for a selected pair, i.e., \({\mathbf {A}}_{i,j}=1\)  when \({\mathbf {S}}_{i,j}\)  is the maximum within \(\lbrace {\mathbf {S}}_{i,j^{\prime }} \mid q_i = d_{j^{\prime }} \rbrace \) .
\(||{\mathbf {A}}||_0 \le n\)  in this case.

